Author Archives: Mats Granvik

Magic series and Magic constants

Craig Knecht sent me an email explaining magic series and magic constants. The following program lists magic series that add up to certain constants using the TableForm command in Mathematica: Mathematica 8: (*program for reordering of integer partitions start*) TableForm[ … Continue reading

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Riemann zeta function on the critical line as a Fourier transform of exponential sawtooth function plus minus one

Riemann zeta function on the critical line as a Fourier transform of exponential sawtooth function plus minus one The code does not work when copy pasted in this blogging platform, so here is a link to Pastebin with some working … Continue reading

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The summation symbol

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A visual interpretation of Riemann zeta zeros via the Fourier transform

Mathematica 8: scale = 1000000; xres = .001; limit = 3000; x = Exp[Range[0, Log[scale], xres]]; a = FourierDCT[(SawtoothWave[x])*x^(-1/2)]; b = -FourierDST[(SawtoothWave[x] – 1)*x^(-1/2)]; (*ListLinePlot[((SawtoothWave[x])*x^(-1/2))[[1;;limit]]]*) gs = ListLinePlot[-((SawtoothWave[x] – 1)*x^(-1/2))[[1 ;; limit]], PlotStyle -> RGBColor[1, 0, 1]]; gsine = ListLinePlot[ … Continue reading

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Illustration of the Discrete Fourier Tranform DFT

Mathematica 8: Do[ nn = i; Print[MatrixForm[Transpose[Table[{1}, {n, 1, nn}]]]] Print[MatrixForm[ Table[Table[Cos[-2*Pi*(n – 1)*(k – 1)/nn], {k, 1, nn}], {n, 1, nn}]]] Print[MatrixForm[ Chop[N[Table[ Total[Table[Cos[-2*Pi*(n – 1)*(k – 1)/nn], {k, 1, nn}]], {n, 1, nn}]]]]] , {i, 1, 12}] Signal … Continue reading

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Dirichlet character sums for the terms of the von Mangoldt function

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The fundamental theorem of arithmetic is encoded by the von Mangoldt function

Mathematica 8 A = Table[ Table[If[Mod[n, k] == 0, Exp[MangoldtLambda[n/k]], “”], {k, 1, 12}], {n, 1, 12}]; MatrixForm[A] Row products of the matrix above are the natural numbers.

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